Taking centre $(0,0)$ and radius $a$, let $M(h,k)$ be the midpoint of the chord of contact $AB$.
Slope of $OM = k/h$, so slope of $AB = -h/k$ (since $AB \perp OM$).
Equation of $AB$: $y - k = -\frac{h}{k}(x-h)$, i.e. $hx + ky = h^2 + k^2$ ... (1)
Equation of tangent at $A(x_1, y_1)$: $xx_1 + yy_1 = a^2$ ... (2)
From (1) and (2): $\frac{x_1}{h} = \frac{y_1}{k} = \frac{a^2}{h^2+k^2}$
$\therefore x_1 = \frac{ha^2}{h^2+k^2},\ y_1 = \frac{ka^2}{h^2+k^2}$.
Since $(x_1, y_1)$ lies on the line $lx_1 + my_1 + n = 0$:
$l \cdot \frac{ha^2}{h^2+k^2} + m \cdot \frac{ka^2}{h^2+k^2} + n = 0$
$\Rightarrow lha^2 + mka^2 + n(h^2+k^2) = 0$
$\Rightarrow h^2 + k^2 + \frac{la^2}{n}h + \frac{ma^2}{n}k = 0$
So the locus of $M$ is the circle $x^2 + y^2 + \frac{la^2}{n}x + \frac{ma^2}{n}y = 0$.